1. 06/16/15 BCM 2 1 Transverse Dynamics - Measurements [MCCPB, Chapter 2] some data analysis techniques coherent oscillations & filamentation

2. 06/16/15 BCM 2 2 Data Analysis Techniques measuring beam response to many steering corrections at many BPMs, adjusting a large number of model parameters in a global fit to best reproduce the data (LOCO code, J. Safranek) singular-value decomposition (SVD); powerful, broad range of applications, e.g. reducing corrector strengths (V. Ziemann, 1992), orthogonal tuning knobs (P. Raimondi, 1998), BPM data analysis (J. Irwin, 1998) principal-axes transformations (‘eigenfit’); may be superior to least-squares fit (T. Lohse, P. Emma, 1989) model-independent analysis (MIA/PCA); extract temporal and spatial patterns from BPM data; identifies relevant physical variables (J. Irwin, C.-X. Wang, Y. Yan, 1998) independent component analysis (ICA); more robust w.r.t. noise, coupling, and nonlinearity; extract transverse betatron phase & amplitude, dispersion, coupling, slices ( X. Huang, 2005; F. Wang, 2008; X. Pang, 2009);

3. 06/16/15 BCM 2 3 orbit response measurements N BPMs reading orbit change  x M dipole correctors exciting deflection angles  closed orbit in a storage ring trajectory Iin part of a ring or transport line

4. 06/16/15 BCM 2 LOCO* – orbit response measurements and analysis 4 *J. Safranek, “Experimental determination of storage ring optics usins closed orbit response measurements”, Nucl. Inst. and Meth. A388, (1997), pg. 27. closed orbit trajectory

5. 06/16/15 BCM 2 LOCO cont’d 5 measured R must be adjusted to match the model; then optical functions are obtained from matched model - in a transfer line only the R matrix can be verified The measured R also depends on the BPM and corrector calibrations: 1) Data preparation: for all i and j A vector holding the weighted difference between the measured and modelled response is built from all matrix elements :

6. 06/16/15 BCM 2 6 LOCO cont’d 2) Local gradient: Evaluate the sensitivity w.r.t. parameters (BPM and corrector calibrations, strengths…). Straightforward for calibrations, requires MADX runs for model parameters (quad strengths…) → linear approximation. 3) Least-square minimization: Solve the linearized equation for parameter changes Δc (based on SVD). 4) Iteration:

7. 06/16/15 BCM 2 CERN accelerator complex 7

8. 06/16/15 BCM 2 matrix sizes 8 For a ring /line with N BPMs and M correctors per plane, the minimum size of the gradient matrix G is : (2 ×N ×M) ×(2 ×(N + M)) …with only BPM and corrector calibrations as parameters for c. SPS transfer line: N < 30, M < 30 1800 x 120 0.2 x 10 6 elements TT10 SPS ring: N ~ 110, M = 108 25000 x 220 6 x 10 6 elements LHC: N ~ 500, M ~ 250 250000 x 1500 375 x 10 6 elements the complete LHC is tough to handle with all elements included: RAM + precision + CPU time

9. 06/16/15 BCM 2 SPS example : before fit 9 J. Wenninger

10. 06/16/15 BCM 2 10 SPS example : a few fit iterations later … J. Wenninger

11. 06/16/15 BCM 2 11 example : TI8 - SPS to LHC transfer line - quadrupole with wrong setting J. Wenninger

12. 06/16/15 BCM 2 12 Singular Value Decomposition The singular value decomposition of a matrix is usually referred to as the SVD. This is the final and best factorization of any m × n X matrix whose entries are real numbers. The SVD factorization is of the form X = U  V t where U is an m × m orthogonal matrix, Σ is an m × n diagonal matrix with non-negative real numbers on the diagonal, and the n × n unitary matrix V t denotes the transpose of the n × n orthogonal matrix V. Such a factorization is called a singular value decomposition of X. The r (nonzero) diagonal entries σ i of Σ are known as the singular values of X. A common convention is to list the singular values in descending order. In this case, the diagonal matrix Σ is uniquely determined by X (though the matrices U and V are not). v 1, v 2,...v r is an orthonormal basis for the row space. u 1, u 2,...u r is an orthonormal basis for the column space v r+1,...v n is an orthonormal basis for the nullspace u r+1,...u m is an orthonormal basis for the left nullspace

13. 06/16/15 BCM 2 13 SVD - continued eigenvector of XX t eigenvector of X t X X = U  V t X -1 = V  -1 U t “pseudo-inverse” where  -1 is a diagonal matrix with elements 1/s 11,… mxmmxm mxnmxn nxnnxn r is the rank of the matrix OR

14. 06/16/15 BCM 2 model-independent analysis (MIA) original idea: no attempt to determine exact parameters of optics model, but use beam information and beam response to certain perturbation to monitor stability, to identify misaligned linac rf structures, etc. m BPMs p measurements form huge matrix

15. 06/16/15 BCM 2 15 decompose matrix with SVD W eigenvectors form basis of temporal patterns V eigenvectors form basis of spatial patterns Singular values computed for the SLAC linac, using 5000 pulses and 130 BPMs only about 6-15 independent variables affecting the beam motion a few high-resolution BPMs BPM noise floor “signals”

16. 16 independent-component analysis (ICA) ICA calculated l source signals, and determines (unknown) mixing matrix A ; source signal independence is related to “diagonality” of covariance matrices time-lagged source signal covariance matrix modified time-lagged co-variance matrix variance apply SVD: where W is the unitary demixing matrix and D k is a diagonal matrix apply SVD:

17. independent-component analysis (ICA) – cont’d 91 *J.F. Cardoso and A. Souloumiac, SIAM J. Matrix Anal. Apl. 17, 161 (1996)

18. Simple description of ICA Despite its wide range of applicability, ICA can be understood in terms of the classic ‘cocktail party’ problem, which ICA solves in an ingenious manner. Consider a cocktail party where many people are talking at the same time. If a microphone is present then its output is a mixture of voices. When given such a mixture, ICA identifies those individual signal components of the mixture that are unrelated. Given that the only unrelated signal components within the signal mixture are the voices of different people, this is precisely what ICA finds. ICA does not incorporate any knowledge specific to speech signals; in order to work, it requires simply that the individual voice signals are unrelated. 18 TRENDS in Cognitive Sciences Vol.6 No.2 February 2002

19. 19 coherent tune shift along a long bunch ICAMIA/PCA X. Pang et al, Phys. Rev. ST - Accel. Beams 15, 112802 (2012)

20. 06/16/15 BCM 2 20 coherent oscillations & filamentation A) response to a kick: coherent damping exp. decay amplitude-dependent oscillation frequency head-tail damping synchrotron-radiation damping bunch population chromaticity

21. 06/16/15 BCM 2 21 1/  1/(100 turns) 1/  SR IbIb Q’=2.7 Q’=14 LEP 45.63 GeV, damping rate 1/  vs. I bunch for different chromaticities [A.-S. Muller] horizontal damping partition number

22. 06/16/15 BCM 2 22 LEP: tune change during damping Q turns [A.-S. Muller]

23. 06/16/15 BCM 2 23 LEP: detuning with amplitude from single kick Q 2 J [A.-S. Muller]

24. 06/16/15 BCM 2 24 B) another response to a kick: filamentation R. Meller et al., SSC-N -360, 1987 both different from e -t/  Z: kick in   =(2  0 ) 2 Q=Q 0 -  a 2 amplitude in  ex.:

25. 06/16/15 BCM 2 25 amplitude decoherence factor vs. turn number 5  kick 1/5  kick oscillation amplitude

26. 06/16/15 BCM 2 26 Horizontal phase space generated with MAD at the pick-up in section 10. Two chains of stable islands (period 4 and 5) are clearly visible. The inner chain is the one studied during the measurement sessions. Upper part: Horizontal beam position vs. time measured with a pick-up for the few turns right after the kick (left) and for the last 1500 turns (right). Bottom part: Normalised phase space reconstructed from two pick-ups. measurement in the CERN PS (A.-S. Muller et al., EPAC 2002) simulated phase space a fraction of particles trapped in the resonance island

27. 06/16/15 BCM 2 27 C) 3 rd response to a kick: decoherence due to chromaticity  >0  <0 (Q’>0) t=0 t=T s /2t=T s

28. 06/16/15 BCM 2 28 R. Meller et al., SSC-N -360, 1987 time TsTs large  small 

29. 06/16/15 BCM 2 29 measurement in the CERN PS (A.-S. Muller at al., EPAC 2002) Measurements of de- and recoherence for two synchrotron frequencies. Both plots are composed of several individual datasets with measurements started at different times after the kick (the current setup only allows the acquisition of about 2000 turns).

30. 06/16/15 BCM 2 Summary data analysis techniques LOCO fits SVD MIA, ICA beam response to kick excitation coherent damping filamentation chromaticity

31. 06/16/15 BCM 2 Computer lab, Wednesday 17 June 2015BCM consider a simple storage ring consisting of a defocusing quadrupole and a focusing quadrupole and two bends (or drifts) compute tune and  at focusing quadrupole compute tune shift as a function of quadrupole strength change  K over a range -0.1…+0.1 compute  as and plot vs.  K compute  asand plot vs  K use L=1 m, K=0.5 m -1 hints: 2cos(2  Q)=R 11 +R 22 R 12 =  sin(2  Q) 1